# Displacement and Equations of Kinematics in Two Dimensions

The following problems and conceptual questions are from chapter 3 of "Physics, 7th edition by Cutnell and Johnson". It pertains to displacement, velocity, and acceleration; equations of kinematics in two dimensions; and projectile velocity.

Please consider significant figures and if possible, draw visual graphics.

Conceptual Questions:

1. An object is thrown upward at an angle θ above the ground, eventually return to earth.

(a) Is there any place along the trajectory where the velocity and acceleration are perpendicular? If so, where?

(b) Is there any place where the velocity and acceleration are parallel? If so, where? In each case, explain.

2. Is the acceleration of a projectile equal to zero when it reaches the top of its trajectory? If not, why not?

3. A wrench is accidentally dropped from the top of the mast of a sailboat. Will the wrench hit at the same place on the deck whether the sailboat is at rest or moving with a constant velocity? Justify your answer.

4. A stone is thrown horizontally from the top of a cliff and eventually hits the ground below. A second stone is dropped from the rest from the same cliff, falls through the same height, and also hits the ground below. Ignore air resistance. Discuss whether each of the following quantities is different or the same in the two cases; if there is a difference, describe the difference: (a) displacement (b) speed just before impact the ground, and (c) time of flight.

5. A leopard springs upward at a 45 degree angle and then falls back to the ground. Does the leopard, at any point on its trajectory, ever have a speed that is one-half its initial value? Give your reasoning.

6. A child is playing on the floor of a recreational vehicle (RV) as it moves along the highway at a constant velocity. He has toy cannon, which shoots a marble at a fixed angle and speed with respect to the floor. The cannon can be aimed toward the front or the rear of the RV. Is the range toward the front the same as, less than, or greater than the range toward the rear? Answer this question (a) from the child's point of view and (b) from the point of view of the observer standing still on the ground. Justify your answer.

7. Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across the river in the least time. Swimmer A swims perpendicular to the current and lands on the far shore downstream, because the current have swept him in that direction. Swimmer B swims upstream at an angle to the current and lands on the far shore directly opposite the starting point. Swimmer C swims downstream at an angle to the current in an attempt to take advantage of the current. Who crosses the river in the least time? Account for your answer.

Problems:

(Displacement, Velocity, and Acceleration)

1. In diving to a depth of 750 m, an elephant seal also moves 460 m due east of his starting point. What is the magnitude of the seal's displacement?

2. In a football game a kicker attempts a field goal. The ball remains in contact with the kicker's foot for 0.050 s, during which time it experiences and acceleration of 340 m/s^2. The ball is launched at an angle of 51 degree above the ground. Determine the horizontal and vertical components of the launch velocity.

3. In a mall, a shopper rides up an escalator between floors. At the top of the escalator, the shopper turns right and walks 9.00 m to a store. The magnitude of the shopper's displacement from the bottom of the escalator is 16.0 m. The vertical distance between the floors is 6.00 m. At what angle is the escalator inclined above the horizontal?

4. The earth moves around the sun in a nearly circular orbit of radius 1.50 x 10^11 m. During the three summer months (an elapsed time of 7.89 x 10^6 s), the earth moves one-fourth of the distance around the sun.

(a) What is the average speed of the earth?

(b) What is the magnitude of the average velocity of the earth during this period?

(Equations of Kinematics in Two Dimensions, and projectile motion)

5. Michael Jordan, formerly the Chicago Bulls basketball team, had some fanatic fans. They claimed that he was able to jump and remain in the air for two full seconds from launch to landing. Evaluate this claim by calculating the maximum height that such a jump would attain. For comparison, Jordan's maximum jump height has been estimated at about one meter.

6. A major-league pitcher can throw a baseball in excess of 41.0 m/s. If a ball is thrown horizontally at this speed, how much will it drop by the time it reaches a catcher who is 17.0 m away from the point of release?

7. A quarterback claims that he can throw the football a horizontal distance of 183 m (200 yard). Furthermore, he claims that he can do this by launching the ball at the relatively low angle of 30.0 degree above the horizontal. To evaluate his claim, determine the speed with which this quarterback must throw the ball. Assume that the ball is launched and caught at the same vertical level and that air resistance can be ignored. For the comparison, a baseball pitcher who can accurately throw a fastball at 45 m/s (100 mph) would be considered exceptional.

8. You are in a hot air-balloon that, relative to the ground, has a velocity of 6.0 m/s in a direction due east. You see a hawk moving directly away from the balloon in a direction due north. The speed of the hawk relative to you is 2.0 m/s. What are the magnitude and direction of the hawk's velocity relative to the ground? Express the directional angle relative to due east.

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#### Solution Preview

Conceptual Questions:

**Please consider significant figures and if possible, draw visual graphics**

1. An object is thrown upward at an angle above the ground, eventually return to earth. (a) Is there any place along the trajectory where the velocity and acceleration are perpendicular? If so, where? (b) Is there any place where the velocity and acceleration are parallel? If so, where? In each case, explain.

Answer:

v

a = g

θ

a) Above fig. shows the trajectory of the object. Velocity vector at any instant is tangential to the trajectory at that instant. Acceleration vector is always vertically down. Hence, velocity and acceleration vectors are perpendicular at the highest point.

b) There is no point where the two vectors are parallel.

2. Is the acceleration of a projectile equal to zero when it reaches the top of its trajectory? If not, why not?

Answer: Acceleration of the projectile is equal to 9.8 m/s2 downwards at every point of the trajectory including the highest point.

3. A wrench is accidentally dropped from the top of the mast of a sailboat. Will the wrench hit at the same place on the deck whether the sailboat is at rest or moving with a constant velocity? Justify your answer.

Answer: The wrench, when dropped has the same horizontal speed as the boat. Also neglecting air friction, this horizontal speed remains constant throughout its downwards motion as there is no force acting on it horizontally. Hence, the horizontal displacement of the wrench and the boat by the time the wrench hits the deck will be same (assuming the boat also moves with constant velocity). The wrench will, therefore, hit the same spot on the deck where it would hit if the boat were at rest.

4. A stone is thrown horizontally from the top of a cliff and eventually hits the ground below. A second stone is dropped from the rest from the same cliff, falls through the same height, and also hits the ground below. Ignore air resistance. Discuss whether each of the following quantities is different or the same in the two cases; if there is a difference, describe the difference: (a) displacement (b) speed just before impact the ground, and (c) time of flight.

Answer: a) Vertical displacement of both stones is same (equal to the height of the cliff). However, horizontal displacements are different at the one falling vertically down falls near the foot of the cliff and the one projected horizontally at some distance. Hence, resultant displacements in the two cases are different.

b) Vertical velocity of each stone will be same on hitting the ground as in each case the initial vertical velocity is zero and vertical displacement is same. However, the stone falling vertically has zero horizontal velocity, whereas the one projected horizontally has a horizontal velocity component at the instant of hitting the ground. Hence, the two stones will have different speeds just before hitting the ground.

c) Vertically both stones cover the same displacement and have same initial vertical velocities (zero). Hence, the time taken to hit the ground (time of flight) will be same in the two cases.

5. A leopard springs upward at a 45o angle and then falls back to the ground. Does the leopard, at any point on its trajectory, ever have a speed that is one-half its initial value? Give your reasoning.

Answer:

...

#### Solution Summary

The displacement and equations of kinematics in two dimensions are examined.

Physics: Density, Vectors, Speed, Acceleration, Velocity

See attached file for proper formatting.

1.7. Which of the following equations are dimensionally correct? (a) vf = vi + ax

(b) y = (2 m)cos(kx), where k = 2 m-1.

1.26. The radius of a solid sphere is measured to be (6.50 ± 0.20) cm, and its mass is measured to be (1.85 ± 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

1. 44. Vector has x and y components of -8.70 cm and 15.0 cm, respectively; vector has x and y components of 13.2 cm and -6.60 cm, respectively. If, what are the components of ?

2. 12. An object moves along the x axis according to the equation x(t) = (3.00t2 - 2.00t + 3.00) m, where t is in seconds. Determine (a) the average speed between t = 2.00 s and t = 3.00 s, (b) the instantaneous speed at t = 2.00 s and at t = 3.00 s, (c) the average acceleration between t = 2.00 s and t = 3.00 s, and (d) the instantaneous acceleration at t = 2.00 s and t = 3.00 s.

2. 25. A truck on a straight road starts from rest, accelerating at 2.00 m/s2 until it reaches a speed of 20.0 m/s. Then the truck travels for 20.0 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.00 s. (a) How long is the truck in motion? (b) What is the average velocity of the truck for the motion described?

2. 55. A rock is dropped from rest into a well. The sound of the splash is heard 2.40 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s.

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